Linkings and Pfaff's equations

Abstract

CONTENTS § I. Introduction. Contact structures 1) Pfaff's generating equations 2) Contact structures 3) Standard examples 4) The isomorphism problem 5) A non-standard example 6) The Hopf bifurcation 7) A survey of results § II. Closed braids, the Reeb foliation, and Markov surfaces 8) Some questions of knot theory. Presentation of a result on closed braids (a lower bound for the genus) 9) The Reeb foliation and Markov surfaces. Existence of Markov surfaces of maximum characteristic 10) Description of Markov surfaces 11) Proof of Markov's theorem on the equivalence of closed braids 12) Proof of a theorem of Magnus and Peluso on unknotted closed braids of three or fewer threads 13) Proof of a lower bound for the genus 14) The question of the Gordian number. First proof of for links of an algebraic singularity 15) Algebraic proof of the formula 16) Interpretation of in contact geometry. A global formula for an analytic complex curve in § III. Linkings of curves and fields of planes 17) The linking coefficient of a link with a field of planes transversal to 18) The self-linking and twistedness of a link in a field of planes tangent to 19) A relation between , , and in the case when is a contact structure 20) The number and the twisted Euler characteristic 21) Another proof of the formula 22) Complex points on a surface in . The number and the Maslov index. A new proof of the formula in 16) § IV. The general idea of contact structures in dimension 3 23) Curves transversal to a standard contact structure on , and closed braids 24) 25) Curves transversal to on 26) Legendre curves of : 27) Implicit differential equations 28) Geometry in elliptic space 29) Contact diffeomorphisms are -incomplete in the space of all diffeomorphisms 30) Examples of non-standard contact structures on a sphere of dimension 3 References